Basic Business Statistics 10th Edition Chapter 3 Numerical Descriptive Measures Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc..

Learning Objectives In this chapter, you learn: To describe the properties of central tendency, variation, and shape in numerical data To calculate descriptive summary measures for a population To calculate the coefficient of variation and Z-scores To construct and interpret a box-and-whisker plot To calculate the covariance and the coefficient of correlation Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Chapter Topics Measures of central tendency, variation, and shape Mean, median, mode, geometric mean Quartiles Range, interquartile range, variance and standard deviation, coefficient of variation, Z-scores Symmetric and skewed distributions Population summary measures Mean, variance, and standard deviation The empirical rule and Chebyshev rule Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Chapter Topics Five number summary and box-and-whisker plot (continued) Five number summary and box-and-whisker plot Covariance and coefficient of correlation Pitfalls in numerical descriptive measures and ethical issues Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Summary Measures Describing Data Numerically Central Tendency Quartiles Variation Shape Arithmetic Mean Range Skewness Median Interquartile Range Mode Variance Geometric Mean Standard Deviation Coefficient of Variation Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Measures of Central Tendency Overview Central Tendency Arithmetic Mean Median Mode Geometric Mean Midpoint of ranked values Most frequently observed value Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Arithmetic Mean The arithmetic mean (mean) is the most common measure of central tendency For a sample of size n: Sample size Observed values Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Arithmetic Mean The most common measure of central tendency (continued) The most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Mean = 3 Mean = 4 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Median In an ordered array, the median is the “middle” number (50% above, 50% below) Not affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Median = 3 Median = 3 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Finding the Median The location of the median: If the number of values is odd, the median is the middle number If the number of values is even, the median is the average of the two middle numbers Note that is not the value of the median, only the position of the median in the ranked data Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Mode A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical (nominal) data There may may be no mode There may be several modes 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 No Mode Mode = 9 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Review Example Five houses on a hill by the beach House Prices: $2,000,000 500,000 300,000 100,000 100,000 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Review Example: Summary Statistics House Prices: $2,000,000 500,000 300,000 100,000 100,000 Sum $3,000,000 Mean: ($3,000,000/5) = $600,000 Median: middle value of ranked data = $300,000 Mode: most frequent value = $100,000 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Which measure of location is the “best”? Mean is generally used, unless extreme values (outliers) exist Then median is often used, since the median is not sensitive to extreme values. Example: Median home prices may be reported for a region – less sensitive to outliers Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Quartiles Quartiles split the ranked data into 4 segments with an equal number of values per segment 25% 25% 25% 25% Q1 Q2 Q3 The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger Q2 is the same as the median (50% are smaller, 50% are larger) Only 25% of the observations are greater than the third quartile Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Quartile Formulas Find a quartile by determining the value in the appropriate position in the ranked data, where First quartile position: Q1 = (n+1)/4 Second quartile position: Q2 = (n+1)/2 (the median position) Third quartile position: Q3 = 3(n+1)/4 where n is the number of observed values Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Quartiles Example: Find the first quartile (n = 9) Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22 (n = 9) Q1 is in the (9+1)/4 = 2.5 position of the ranked data so use the value half way between the 2nd and 3rd values, so Q1 = 12.5 Q1 and Q3 are measures of noncentral location Q2 = median, a measure of central tendency Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Quartiles Example: (n = 9) (continued) Example: Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22 (n = 9) Q1 is in the (9+1)/4 = 2.5 position of the ranked data, so Q1 = 12.5 Q2 is in the (9+1)/2 = 5th position of the ranked data, so Q2 = median = 16 Q3 is in the 3(9+1)/4 = 7.5 position of the ranked data, so Q3 = 19.5 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Geometric Mean Geometric mean Geometric mean rate of return Used to measure the rate of change of a variable over time Geometric mean rate of return Measures the status of an investment over time Where Ri is the rate of return in time period i Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Example An investment of $100,000 declined to $50,000 at the end of year one and rebounded to $100,000 at end of year two: 50% decrease 100% increase The overall two-year return is zero, since it started and ended at the same level. Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Example (continued) Use the 1-year returns to compute the arithmetic mean and the geometric mean: Arithmetic mean rate of return: Misleading result Geometric mean rate of return: More accurate result Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Coefficient of Variation Measures of Variation Variation Range Interquartile Range Variance Standard Deviation Coefficient of Variation Measures of variation give information on the spread or variability of the data values. Same center, different variation Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Range = Xlargest – Xsmallest Simplest measure of variation Difference between the largest and the smallest values in a set of data: Range = Xlargest – Xsmallest Example: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Range = 14 - 1 = 13 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Disadvantages of the Range Ignores the way in which data are distributed Sensitive to outliers 7 8 9 10 11 12 7 8 9 10 11 12 Range = 12 - 7 = 5 Range = 12 - 7 = 5 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5 Range = 5 - 1 = 4 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120 Range = 120 - 1 = 119 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Interquartile Range Can eliminate some outlier problems by using the interquartile range Eliminate some high- and low-valued observations and calculate the range from the remaining values Interquartile range = 3rd quartile – 1st quartile = Q3 – Q1 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Interquartile Range Example: Median (Q2) X X Q1 Q3 Interquartile range maximum minimum 25% 25% 25% 25% 12 30 45 57 70 Interquartile range = 57 – 30 = 27 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Variance Average (approximately) of squared deviations of values from the mean Sample variance: Where = mean n = sample size Xi = ith value of the variable X Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Standard Deviation Most commonly used measure of variation Shows variation about the mean Is the square root of the variance Has the same units as the original data Sample standard deviation: Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Calculation Example: Sample Standard Deviation Sample Data (Xi) : 10 12 14 15 17 18 18 24 n = 8 Mean = X = 16 A measure of the “average” scatter around the mean Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Measuring variation Small standard deviation Large standard deviation Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Comparing Standard Deviations Data A Mean = 15.5 S = 3.338 11 12 13 14 15 16 17 18 19 20 21 Data B Mean = 15.5 S = 0.926 11 12 13 14 15 16 17 18 19 20 21 Data C Mean = 15.5 S = 4.567 11 12 13 14 15 16 17 18 19 20 21 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Advantages of Variance and Standard Deviation Each value in the data set is used in the calculation Values far from the mean are given extra weight (because deviations from the mean are squared) Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Coefficient of Variation Measures relative variation Always in percentage (%) Shows variation relative to mean Can be used to compare two or more sets of data measured in different units Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Comparing Coefficient of Variation Stock A: Average price last year = $50 Standard deviation = $5 Stock B: Average price last year = $100 Both stocks have the same standard deviation, but stock B is less variable relative to its price Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Z Scores A measure of distance from the mean (for example, a Z-score of 2.0 means that a value is 2.0 standard deviations from the mean) The difference between a value and the mean, divided by the standard deviation A Z score above 3.0 or below -3.0 is considered an outlier Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Z Scores (continued) Example: If the mean is 14.0 and the standard deviation is 3.0, what is the Z score for the value 18.5? The value 18.5 is 1.5 standard deviations above the mean (A negative Z-score would mean that a value is less than the mean) Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Shape of a Distribution Describes how data are distributed Measures of shape Symmetric or skewed Left-Skewed Symmetric Right-Skewed Mean < Median Mean = Median Median < Mean Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Using Microsoft Excel Descriptive Statistics can be obtained from Microsoft® Excel Use menu choice: tools / data analysis / descriptive statistics Enter details in dialog box Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Using Excel Use menu choice: tools / data analysis / descriptive statistics Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Using Excel Enter dialog box details Check box for summary statistics (continued) Enter dialog box details Check box for summary statistics Click OK Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Excel output Microsoft Excel descriptive statistics output, using the house price data: House Prices: $2,000,000 500,000 300,000 100,000 100,000 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Numerical Measures for a Population Population summary measures are called parameters The population mean is the sum of the values in the population divided by the population size, N Where μ = population mean N = population size Xi = ith value of the variable X Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Population Variance Average of squared deviations of values from the mean Population variance: Where μ = population mean N = population size Xi = ith value of the variable X Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Population Standard Deviation Most commonly used measure of variation Shows variation about the mean Is the square root of the population variance Has the same units as the original data Population standard deviation: Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

The Empirical Rule If the data distribution is approximately bell-shaped, then the interval: contains about 68% of the values in the population or the sample 68% Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

The Empirical Rule contains about 95% of the values in the population or the sample contains about 99.7% of the values in the population or the sample 95% 99.7% Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Chebyshev Rule Regardless of how the data are distributed, at least (1 - 1/k2) x 100% of the values will fall within k standard deviations of the mean (for k > 1) Examples: (1 - 1/12) x 100% = 0% ……..... k=1 (μ ± 1σ) (1 - 1/22) x 100% = 75% …........ k=2 (μ ± 2σ) (1 - 1/32) x 100% = 89% ………. k=3 (μ ± 3σ) At least within Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Approximating the Mean from a Frequency Distribution Sometimes only a frequency distribution is available, not the raw data Use the midpoint of a class interval to approximate the values in that class Where n = number of values or sample size c = number of classes in the frequency distribution mj = midpoint of the jth class fj = number of values in the jth class Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Approximating the Standard Deviation from a Frequency Distribution Assume that all values within each class interval are located at the midpoint of the class Approximation for the standard deviation from a frequency distribution: Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Exploratory Data Analysis Box-and-Whisker Plot: A Graphical display of data using 5-number summary: Minimum -- Q1 -- Median -- Q3 -- Maximum Example: 25% 25% 25% 25% Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Shape of Box-and-Whisker Plots The Box and central line are centered between the endpoints if data are symmetric around the median A Box-and-Whisker plot can be shown in either vertical or horizontal format Min Q1 Median Q3 Max Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Distribution Shape and Box-and-Whisker Plot Left-Skewed Symmetric Right-Skewed Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Box-and-Whisker Plot Example Below is a Box-and-Whisker plot for the following data: 0 2 2 2 3 3 4 5 5 10 27 The data are right skewed, as the plot depicts Min Q1 Q2 Q3 Max 0 2 3 5 27 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

The Sample Covariance The sample covariance measures the strength of the linear relationship between two variables (called bivariate data) The sample covariance: Only concerned with the strength of the relationship No causal effect is implied Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Interpreting Covariance Covariance between two random variables: cov(X,Y) > 0 X and Y tend to move in the same direction cov(X,Y) < 0 X and Y tend to move in opposite directions cov(X,Y) = 0 X and Y are independent Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Coefficient of Correlation Measures the relative strength of the linear relationship between two variables Sample coefficient of correlation: where Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Features of Correlation Coefficient, r Unit free Ranges between –1 and 1 The closer to –1, the stronger the negative linear relationship The closer to 1, the stronger the positive linear relationship The closer to 0, the weaker the linear relationship Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Scatter Plots of Data with Various Correlation Coefficients Y Y Y X X X r = -1 r = -.6 r = 0 Y Y Y X X X r = +1 r = +.3 r = 0 Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Using Excel to Find the Correlation Coefficient Select Tools/Data Analysis Choose Correlation from the selection menu Click OK . . . Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Using Excel to Find the Correlation Coefficient (continued) Input data range and select appropriate options Click OK to get output Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Interpreting the Result There is a relatively strong positive linear relationship between test score #1 and test score #2 Students who scored high on the first test tended to score high on second test, and students who scored low on the first test tended to score low on the second test Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Pitfalls in Numerical Descriptive Measures Data analysis is objective Should report the summary measures that best meet the assumptions about the data set Data interpretation is subjective Should be done in fair, neutral and clear manner Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Ethical Considerations Numerical descriptive measures: Should document both good and bad results Should be presented in a fair, objective and neutral manner Should not use inappropriate summary measures to distort facts Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Chapter Summary Described measures of central tendency Mean, median, mode, geometric mean Discussed quartiles Described measures of variation Range, interquartile range, variance and standard deviation, coefficient of variation, Z-scores Illustrated shape of distribution Symmetric, skewed, box-and-whisker plots Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.

Chapter Summary Discussed covariance and correlation coefficient (continued) Discussed covariance and correlation coefficient Addressed pitfalls in numerical descriptive measures and ethical considerations Basic Business Statistics, 10e © 2006 Prentice-Hall, Inc.